XVII - Hypothesis Testing for the mean - SLC Home Page

MATHEMATICS 201-510-LW
Business Statistics
Martin Huard
Fall 2008
XVII – Hypothesis Testing for µ (large samples)
1. Write the null and alternative hypotheses for each of the following examples. Determine if
each is a case of a two-tailed, a left-tailed, or a right-tailed test.
a) To test whether or not the mean price of houses in Quebec is greater than $143000.
b) To test if the mean number of hours spent working per week by college students who
hold jobs is different from 15 hours.
c) To test whether the mean life of a particular brand of auto batteries is less than 45
months.
d) To test if the mean amount of time spent doing homework by all fourth-graders is
different from 5 hours a week.
e) To test if the mean age of all college students is different from 18 years.
2. Consider H : 20
0
µ = versus H
A
: µ < 20
a) What type of error would you make if the null hypothesis is actually false and you fail
to reject it
b) What type of error would you make if the null hypothesis is actually true and you
reject it
3. According to a Pharmacist, Quebecers spent an average of $220 per person on prescription
drugs in 2006. A recent survey of 300 randomly chosen Quebecers showed that they spent
an average of $235 per person on prescription drugs with a standard deviation of $90. Test at
the 2.5% significance level whether the mean amount currently spent on prescription drugs
by all Quebecers exceeds $220 per person. Use the classical approach.
4. According to Statistics Canada, the average family income in Canada was $76 100 in 2004.
A recently taken sample of 1200 Canadian families yielded a mean income of $77 152 with a
sample standard deviation of $16 850. Using the 2% significance level, can you conclude
that the mean family income in Canada has changed since 2004 Use the classical approach.
5. A study conducted a few years ago claims that adult males spend an average of 11 hours a
week watching sports on television. A recent sample of 100 adult males showed that the
mean time they spend per week watching television is 9.50 hours with a standard deviation of
2.2 hours. Test at the 1% significance level if currently all adult males spend less than 11
hours watching sports on television. Use the classical approach.

Math 510
XVII – Hypothesis testing for µ (large samples)
6. A telephone company claims that the mean duration of all long-distance phone calls made by
its residential customers is 10 minutes. A random of 100 long-distance calls made by its
residential customers taken from the records of this company showed that the mean duration
of calls for this sample is 9.0 minutes with a standard deviation of 5.2 minutes. Test the
whether the mean duration of all long-distance calls is less than 10 minutes
(a) at the 2% level of significance
(b) at the 5% level of significance
Use the p-value approach.
7. For a population of humans to sustain itself, there must be an average of just of over two
births for each woman of reproductive age. The fertility rate varies substantially from within
one nation to the next. A journalist recently pointed out in Fortune magazine that the
average for Japan has dropped to 1.5 births for each woman of reproductive age, which could
reduce Japan’s population from 135 million in 1997 to 50 million by the end of the 21 st
century. Suppose a random sample of 200 Japanese women of reproductive age is taken in
2002, and the sample mean fertility rate is measured as 1.45 with a standard deviation of
0.75. Did the rate decline Use a 5% level of significance, with
a) the classical approach
b) the p-value approach
8. A politician claims that the mean number of hours Canadians worked per week in a typical
week is greater than 40 hours. A sample of 324 Canadians workers produced a mean of 41.3
hours per week with a standard deviation of 10.63 hours per week. Test the politician’s
claim at the 5% level of significance using
a) the classical approach
b) the p-value approach
9. The customers at a bank complained about long lines and the time they had to spend waiting
for service. It is known that the customers at this bank had to wait 8 minutes, on average,
before being served. The management made some changes to reduce the waiting time for its
customers. A sample of 32 customers taken after these changes were made produced a mean
waiting time of 7.4 minutes with a standard deviation of 2.1 minutes. Using this sample
mean, the bank manager displayed a huge banner inside the bank mentioning that the mean
waiting time for customer has been reduced by new changes. Do you think the manager’s
claim is justifiable Use a 2.5% level of significance along with the classical approach.
10. A sample of 225 parents were asked how much time they spent per week on school work or
school-related activities. This sample produced a mean of 5.6 hours per week, with a
standard deviation of 4.4 hours. At the 0.01 level of significance, test the claim that the mean
number of hours spent by parents on school work or school-related activities is 5 hours per
week, using
a) the classical approach
b) the p-value approach
Fall 2008 Martin Huard 2

Math 510
XVII – Hypothesis testing for µ (large samples)
ANSWERS
1. a) H : $143000
0
µ =
H
A
: µ > $143000
c) H0 : µ = 45 months
H : µ < 45 months
A
e) H0 : µ = 18 years Two-tailed
H
A
: µ ≠ 18 years
2. a) Type II error b) Type I error
3. H : $220
0
µ = critical value: z = 1.96
H
A
: µ > $220
4. H0 : µ = $76 100
H : µ ≠ $76 100
A
5. H0 : µ = 11 hours
H : µ < 11 hours
A
6. H0 : µ = 10 minutes
H : µ < 10 minutes
A
A
Right-tailed b) H0 : µ = 15 hours
H : µ ≠ 15 hours
Left-tailed d) H : 5 hours
0
µ =
0.025
test statistic: z = 2.89
critical values: ± z = ± 2.33
0.01
test statistic: z = 2.16
critical value: − z = −2.33
0.025
test statistic: z = −6.82
H
A
A
Two-tailed
Two-tailed
: µ ≠ 5 hours
Reject H 0
Fail to reject H 0
Reject H 0
test statistic: z = −1.92
a) Fail to reject H
p − value = 0.0274
b) Reject H
7. H0 : µ = 1.5 births critical value: − z0.05
= −1.65
Fail to reject H 0
H
A
: µ < 1.5 births test statistic: z = −0.94
p − value = 0.1736
8. H0 : µ = 40 hours critical value: z0.05
= 1.65
Reject H 0
H : µ > 40 hours test statistic: z = 2.20
p − value = 0.0139
9. H : 8 minutes
0
µ =
critical value: − z = −1.96
H
A
: µ < 8 minutes
10. H : 5 hours
0
µ =
H
A
: µ ≠ 5 hours
0.025
test statistic: z = −1.62
critical values: ± z = ± 2.58
0.005
test statistic: z = 2.05
p − value = 0.0404
Fail to reject H 0
Fail to reject H 0
o
o
Fall 2008 Martin Huard 3